Final answer:
It is true that if two lines are parallel to the same line, they are parallel to each other. Waves' amplitudes combine only when they are co-linear, and the Pythagorean theorem can be used for vectors at right angles.
Step-by-step explanation:
The original question posed is whether two lines that are parallel to the same line are parallel to each other. This is true. This concept is known as the transitive property of parallel lines in geometry. If line A is parallel to line B, and line B is parallel to line C, then by this property, line A must be parallel to line C as well.
For waves, two statements were given relating to amplitude and alignment. First, the statement that the amplitudes of waves add up only if they are propagating in the same line is true. This is because waves can only superimpose and create constructive or destructive interference when they are co-linear, hence modify each other's amplitude. If they are not co-linear, they do not interact in this manner. The second statement about the amplitude being affected only when the waves are precisely aligned is also true, for the same reason.
For vectors, the Pythagorean theorem can be used to calculate the length of the resultant vector when two vectors are at right angles to each other. This is another true statement, as the Pythagorean theorem applies to right-angled triangles, and vectors can indeed form right-angle triangles with their components.